basisobservability

Basis observability refers to the property of a dynamical system indicating whether the state coordinates expressed in a chosen basis can be inferred from output measurements over time. In a linear time-invariant system with ẋ = Ax + Bu and y = Cx + Du, selecting a basis B for the state space and transforming x = Tb yields a transformed system in coordinates b. The basis is observable if the coordinates b can be uniquely determined from y(t) for t ≥ 0, i.e., the transformed system is observable (the observability matrix of the pair (A′, C′) has full rank n). Because observability is invariant under non-singular coordinate changes, a system that is observable in its original coordinates remains observable in any full-rank basis, but certain bases can illuminate observable versus unobservable directions more clearly.

A practical way to exploit basis observability is via the Kalman decomposition, which splits the state space into observable and unobservable subspaces and provides a basis that separates these components. In this basis, the system matrices take a block form that makes it easier to design observers or controllers using only the observable coordinates. Such a basis can also facilitate model reduction by focusing on the observable part of the state.

Nonlinear extensions use local observability and differential-geometric tools, such as Lie derivatives, to assess whether local state reconstruction is possible in a given coordinate chart. Basis observability in nonlinear systems can vary with the operating point and the chosen coordinate representation.

Applications include state estimation, sensor placement, and model reduction, where aligning the basis with the observable subspace improves interpretability and computational efficiency. Limitations arise from modeling errors, measurement noise, and the inherent nonlinearity of real systems.